On equations of finite fields of characteristic 2 and APN functions

Abstract

Let F be the finite field GF(2n) of characteristic 2 and f a quadratic APN function on F. We construct a n(n−3)2 dimensional subspace Wf of F∧F, where F∧F is the alternative product of F, that is, the quotient space of the tensor product F⊗F of F by the subspace 〈x⊗x∣x∈F〉.We denote by S the set of all subspaces Wf constructed from quadratic APN functions f on F. We prove that a group G isomorphic to GL(n,2) acts on S and that there exists a one-to-one correspondence between the extended affine equivalence classes of quadratic APN functions and the set of G-orbits on S. The correspondence above was first observed by Yoshiara in Section 5 of Yoshiara (2010) and Edel (2011) and the author Nakagawa (2009).Moreover we prove F∧F is isomorphic to Fn−12 (for n odd) or Fn2−1×GF(2n2) (for n even) through an explicit isomorphism, and we give practical forms of those subspaces which correspond to the Gold function f(x)=x2k+1 where gcd(n,k)=1 and to the function f(x)=x3+Tr(x9), viewed as subspaces of Fn−12 (for n odd) or Fn2−1×GF(2n2) (for n even) (see Section 6 in Yoshiara, 2010).We estimate the number of solutions of linear equations x2e+1+αx2e+βx2+(α+β+1)x=0 on GF(22e), and then construct some quadratic APN functions.